how can i work out the number of 3*3sudoku rectangles?

Question

i've already worked out it by computer,but i want 2 do it by hand

Answer 1

I answered this first (below) for 9 × 9 boards. For 3 × 3 instead of 9 × 9, it would be the same idea.

3! choices for the first row, 2! for the second, 1! for the third.

3 × 2 × 1 ×
2 × 1 × 1 ×
1 × 1 × 1
-----------
12 possible 3 × 3 sudoku boards.

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Starting with the first row, you have:

9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 9! choices.

Then for the first box in the second row, you only have 8 choices, because you already picked one for the upper left. And then 7 choices for the next, etc.

8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 × 1 = 8! choices
7 × 6 × 5 × 4 × 3 × 2 × 1 × 1 × 1 = 7! choices...

See the pattern?

So I think there should be:

9!8!7!6!5!4!3!2!1! = 1,834,933,472,251,084,800,000 possible sudoku configurations.